PS3 and PSP downloadable software question

Besides just simply playing some of the games, my experience with PS3 and PSP is fairly limited, mostly interface-wise so I'm looking to see if someone could give me a hand..

Are PSN accounts console specific or could I have the same account on both a PS3 and PSP? Does that even make sense?

I guess what I'm asking is, if I downloaded a PSN game on PS3 and then later decided I wanted to play it on the go on PSP (or vice versa), could I transfer one of the downloaded games from one console to the other (assuming it's a PS1 classic or whatever available for both), or, be able to redownload the game for free on the other console? I've heard of remote play but does that require both systems to be running and in close proximity to one another?

I'm not even sure if any of this makes sense because I really have no sweet clue how it all works. Thanks for any help!

Are PSN accounts console specific or could I have the same account on both a PS3 and PSP? You can have the same account

I guess what I'm asking is, if I downloaded a PSN game on PS3 and then later decided I wanted to play it on the go on PSP (or vice versa), could I transfer one of the downloaded games from one console to the other (assuming it's a PS1 ****c or whatever available for both), or, be able to redownload the game for free on the other console? PS1 games and Minis are playable on both the PS3 and PSP

I've heard of remote play but does that require both systems to be running and in close proximity to one another? Yes/No. Yes you have to have both running and with basic setup, the promixity would be the same as a wireless router (Typically anywhere in the house). Now you can set it up so that you could turn on and access your PS3 from anywhere your PSP can connect to the internet.